Square Root of 563

√563 at a glance

Exact value

√563

Decimal (10 places)

23.7276210354

Rounded

23.7 · 23.73 · 23.728

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.727621

Prime factorization

563

Cube root

8.257263

How to simplify √563

563 is a prime number, so its only factors are 1 and 563. There is no perfect-square factor to pull out, which means √563 is already in its simplest radical form.

The square root of any prime is irrational. If √563 were a fraction a/b in lowest terms, then a² = 563b², so 563 would divide a — and then 563 would divide b too, contradicting “lowest terms.” That is why the decimal 23.7276210354 is only a rounded value.

Where √563 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √563 lies between 23 and 24. 563 is 34 above 529 and 13 below 576, so the root is closer to 24.

• Straight line between 529 and 576: 23.7234 (0.02% low)
• Tangent from 23, i.e. 23 + 34 ÷ 46: 23.7391 (0.05% high)
• Tangent from 24, i.e. 24 − 13 ÷ 48: 23.7292 (0.01% high)

For √563 the tangent at 24 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 563 is just 13 below 576.

Finding √563 with the Babylonian method

If a guess is too big, 563 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√563) in one step.

Start from the nearest whole number, 24 (24² = 576):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √563 = 23.7276210354 to every decimal shown.

√563 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √563 the pattern is [23; 1, 2, 1, 2, 23, 2, 1, 2, 1, 46] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √563 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 563y² = 1. Its smallest solution in positive whole numbers is x = 68,122, y = 2,871.

√563 in geometry and everyday measurements

• A square garage floor of 563 square feet measures about 23.73 ft (23 ft 9 in) per side, and its corner-to-corner diagonal is √1126 ≈ 33.6 ft.
• 563 is not a sum of two whole-number squares — 563 is itself a prime that is one less than a multiple of 4, which rules that out — so √563 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 11 × 21 box, because 1² + 11² + 21² = 563.

Square roots near √563 and related roots

• The cube root of 563 is about 8.257263.
• Squaring undoes the root: (√563)² = 563, while 563² = 316,969 — the number whose square root is 563.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.